Yay a science-y thread. What happens in Schrodinger's cat can be applied to the 2nd law of thermodynamics and the statement of Clausius.
In any closed process the 2nd law looks like:
Where
Q = heat transfer
T = temperature
S = entropy
We assume that Schrodinger's cat is an isolated system that is doing something so we can then say.
This means that the entropy of the box that the cat is in will be either increasing or remaining constant. All of this is just what Clausius did. To make this simple we are going to assume the cat doesn't move, breathe, shed, blink or do anything to change the entropy of the box and nothing comes along to interferes with it either and the box only contains the cat and the atom (and stuff to kill the cat). There will be a vacuum everywhere else inside the box. This is so that the only way the entropy of what's inside the box can change is by the atom decaying and eventually killing the cat.
The 2nd law isn't very black and white when applied to single atoms or at any time where things from quantum mechanics are significant compared to macroscopic things like gravity. Simply put, when there are very few atoms the chances that the 2nd law will not be accurate becomes high.
Now then, the instant we close the box we don't really know whats going on in there. At any instant after the box has closed the atom could have decayed and we have a dead cat OR we have a alive cat wondering why the hell its next to some poison but not both.
Gibbs' paradox is similar to this. I won't write it out but the
idea is the same. The end result is that the fact that we opened the box and took a look inside at some instant will increase the entropy of the contents of the box.
Also in Gibbs' paradox no macroscopic changes occur when the gases are allowed to mix. Relating that to the cat, once we close the box we can't notice when the cat dies if it dies. So until we re-open the box (Gibbs': re-insert the divider) the particles inside the box are essentially indistinguishable.
On the scale of the universe things get really messy. Since we are in the view of the cat, we know whether the "atom" has decayed or not, that is we are able to distinguish macroscopic things but if we say our universe is the like the box and our entropy is constantly increasing something has to be "opening our box" to make that increase, meaning to the "observer" we are completely indistinguishable when they are not looking.
Random Thought:
Our universe has so many atoms that we standing here cannot possibly tell the difference between two atoms next to each other at the other end of the universe. We already know that by looking as far away as we can in the universe we are looking into the past of that region. I think that for this very very far away region we can call that region another box with a cat inside and we are not looking
at this instant but looking at some time very very very long ago. That is by us looking very far away we are distinguishing that region as being either "dead" or "alive" but not both. At any instant in the future we are unable to tell the difference because light has not made it from there to here, so essentially we are not looking.
This idea applies to us as well. There are distant "observers" that are seeing us as either "dead" or "alive". That alone is causing our entropy to continuously increase. So in general, for any isolated system (or universe) with more atoms than anyone cares to count something like this happens, where the universe "observes" its own states. Mathematically I have no idea how to describe this.
Enjoyed writing this. We need more science threads.